In a large bowl, whisk together the warm water and yeast. Let it sit for about 5 minutes, until the yeast dissolves and the mixture looks slightly bubbly.
Add the red velvet cake mix and flour, then stir vigorously for about 1 minute, just until the dough comes together and no dry flour remains. Do not knead or overwork the dough — it should be soft and slightly sticky. Cover the bowl with plastic wrap and let the dough rise in a warm, draft-free place until doubled in size, about 1 to 1 hour and 20 minutes.
While the dough is rising, whisk together the brown sugar, cinnamon, and 1/4 teaspoon salt in a medium bowl, breaking up any lumps. Work in 12 tablespoons of the butter until the mixture is fully combined and spreadable. Set aside.
In another bowl, beat together the cream cheese, confectioners’ sugar, heavy cream, vanilla, the remaining 4 tablespoons of butter, and 1/4 teaspoon salt until light and creamy, about 2 minutes. Set the frosting aside.
Once the dough has risen, gently deflate it by lifting around the edges and letting it fall back into the bowl. Transfer the dough to a lightly floured surface and roll it into a 12-by-16-inch rectangle, gently pushing the edges in to keep the sides straight.
Spread the brown sugar filling evenly over the dough, leaving a 1/4-inch border around the edges. Starting from one of the short sides, roll the dough tightly into a log, pressing the sides as needed to keep it even.
Grease a 9-by-13-inch baking dish with butter. Cut the dough log into 12 equal pieces and arrange them cut-side up in the pan, gently reshaping the rolls if needed. Cover and let rise in a warm, draft-free place until the rolls nearly fill the pan, about 30 minutes.
Preheat the oven to 350°F with a rack in the center position. Bake the rolls until puffed, brick red, and bubbling, about 20 to 25 minutes. Let cool for about 30 minutes, until set.
Spread the frosting over the warm rolls and serve.